Warm Up - 3 mins goes in notebook every day
Original: p → q Converse: q → p
Inverse: ~p → ~q Contrapositive: ~q → ~p
~ means “negation” or “not”
Write the converse, inverse, and contrapositive of the following statement:
“If your grade is 96%, then you get an A.”
Deductive Reasoning
I Can determine if a statement is valid and why
Using Logic! or something like it...
Deductive Reasoning
When Alice meets the Pigeon in Wonderland, the Pigeon thinks she is a serpent. The pigeon reasons that serpents eat eggs, and Alice confirms that she has eaten eggs.
Alice: But I'm NOT a serpent, I tell you! I'm a -- I'm a --
The Pigeon: Well! WHAT are you? I can see you're trying to invent something!
Alice: I -- I'm a little girl.
The Pigeon: A likely story indeed! I've seen a good many little girls in my time, but never ONE with such a neck as that! No, no! You're a serpent; and there's no use denying it. I suppose you'll be telling me next that you never tasted an egg!
The Two Laws of Deductive Reasoning
Law of Detachment
If p → q is a true statement and p is true, then q is true.
Law of Syllogism
If p → q and q → r are true statements, then p → r is a true statement.
Graphic Organizer
Using the Law of Detachment
Determine if the conjecture is valid by the Law of Detachment.
Given: If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area of the triangle is 30 cm². The area of ∆PQR is 30 cm².
Conjecture: The side lengths of ∆PQR are 5 cm, 12 cm, and 13 cm.
By the Law of Detachment: If p → q is a true statement and p is true, then q is true.
p – If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm
q – then the area of the triangle is 30 cm²
q – the area of ∆PQR is 30 cm²
*The statement does not follow the logic for the Law of Detachment. The dimensions of the triangle could be different. The base could be 6 cm and the height could be 10 cm. This one goes “If p then q and q is true” which is not the logical ordering for the Law of Detachment.
Using the Law of Detachment
Determine if the conjecture is valid by the Law of Detachment.
Given: If a student passes his classes, then the student is eligible to play sports. Ross passed his classes.
Conjecture: Ross is eligible to play sports.
By the Law of Detachment: If p → q is a true statement and p is true, then q is true.
p – If a student passes his classes
q – then the student is eligible to play sports
p – Ross passed his classes
*The statement does follow the logic of the Law of Detachment. The conditional statement is true and the statement that Ross passed his classes matches the hypothesis. Therefore, the conjecture is valid.
Using the Law of Syllogism
Determine if the conjecture is valid by Law of Syllogism.
Given: If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon.
Conjecture: If a figure is a kite, then it is a polygon.
By the Law of Syllogism: If p → q and q → r are true statements, then p → r is a true statement.
p – a figure is a kite
q – the figure is a quadrilateral
r – the figure is a polygon
*Since a kite is a quadrilateral and a quadrilateral is a polygon, then by Law of Syllogism the kite is a polygon. It does follow the “if p then q and if q then r, therefore if p then r” ordering for Syllogism.
Using the Law of Syllogism
Determine if the conjecture is valid by Law of Syllogism.
Given: If a number is divisible by 2, then it is even. If a number is even, then it is an integer.
Conjecture: If a number is an integer, then it is divisible by 2.
By the Law of Syllogism: If p → q and q → r are true statements, then p → r is a true statement.
p – a number is divisible by 2
q – the number is even
r – the number is an integer
*The ordering of this problems follows the pattern of “If p then q and q then r are true, then r then p.” This does not follow the order for the Law of Syllogism, therefore it is not a valid conclusion to make. You may also reason that not all integers are divisible by 2 (ex. 3 is an integer and it is not divisible by 2).
Using Deductive Reasoning
If two angles are congruent, then they have the same measure. .
Conjecture:
Hypothesis:
The conjecture is (valid/invalid) _________________ by the Law of
_________________.
m<BAC = m <CAD
The two angles are congruent
valid
Detachment
Using Deductive Reasoning
Conjecture:
p –
q –
r –
The conjecture is (valid/invalid) _________________ by the Law of
_________________.
valid
Syllogism
I Can determine if a statement is valid and why
Use the following given information to determine if the conjecture is true. If it’s true, state the law of deductive reasoning that you used (syllogism or detachment).
1) Given: If n is a natural* number, then n is an integer. If n is an integer, then n is a rational number. 0.875 is a rational number.
Conjecture: 0.875 is a natural number.
*Natural numbers are your counting numbers
2) Given: If an American citizen is at least 18 years old, then he or she is eligible to vote. Anna is a 20-year-old American citizen.
Conjecture: Anna is eligible to vote.